## Principle Components of an EBSD System

Typically an EBSD system (Figure 1) consists of:

- A crystalline sample tilted to 70º from horizontal, by use of the SEM stage or a pretilted holder.
- A phosphor screen, which is fluoresced by the electrons scattered from the sample.
- A sensitive camera together with optics for viewing the pattern formed on the phosphor screen.
- An insertion mechanism, which accurately controls the position of the detector when it is in use, and retracts the detector to a safe position when it’s not in use to prevent interference with SEM operation.
- Electronics to control the SEM, including the beam and stage movements.
- A computer and software to control the EBSD experiments, collect and analyse the diffraction patterns, and display the results.
- forescatter diodes (FSD) mounted around the phosphor screen, used to generate microstructure images of the sample before collecting EBSD data.
- The EBSD system can optionally be integrated with an EDS system

## Pattern Formation

The following model describes the principle features of pattern formation and collection for EBSD analysis. A beam of electrons is directed at a point of interest on a tilted crystalline sample. The atoms in the material inelastically scatter a fraction of the electrons, with a small loss of energy, to form a divergent source of electrons close to the surface of the sample. Some of these electrons are incident on atomic planes at angles which satisfy the Bragg equation:

where n is an integer, λ is the wavelength of the electrons, d is the spacing of the diffracting plane, and θ the angle of incidence of the electrons on the diffracting plane.

These electrons are diffracted to form a set of paired large-angle cones that correspond to each diffracting plane. The image produced on the phosphor screen contains characteristic Kikuchi bands which are formed where the regions of enhanced electron intensity intersect the screen (Figure 1). The pattern seen is a gnomonic projection of the diffracted cone, making the band edges appear hyperbolic.

### Band intensity

The mechanisms giving rise to the Kikuchi band intensities and profile shapes are complex. As an approximation, the intensity of a Kikuchi band for the plane (hkl) is given by:

where fi(θ) is the atomic scattering factor for electrons and ( xi yi zi ) are the fractional coordinates in the unit cell for atom i. A collected diffraction pattern should be compared with a simulated pattern calculated using this equation. This ensures only planes that produce visible Kikuchi bands are used when solving the diffraction pattern.

## Basics of Automated Indexing

Once an EBSD system has been calibrated, it is possible to automatically index the diffraction patterns and calculate the crystal orientation. This is typically accomplished using the following steps:

- The diffraction pattern is transferred from the detector to the EBSD software.
- A Hough transform is used to identify the positions of the Kikuchi bands. The bands are seen as peaks in Hough space.
- Having identified the Kikuchi band positions and from knowing the calibrated geometry, it is possible to calculate the angles between the detected bands.
- The calculated angles are compared with a list of interplanar angles for the analysed structure(s) based on a selected number of reflecting planes (reflectors) in the reference structure.
- The possible solutions are sorted to find the best fit and the orientation matrix is calculated.

This whole process is automatic and takes less than a few milliseconds on modern computers.

### The Hough transform

The Hough transform, which identifies the positions of the Kikuchi bands, converts the image from the EBSD camera into a representation in Hough space, by using the following relation between the points (x, y) in the diffraction pattern and the coordinates (ρ, θ) of the Hough space: ρ = *x cosθ + y sinθ. *A straight line in the image space (x, y) can be characterised by ρ, the perpendicular distance from the line to origin and θ, the angle made with the x-axis, and can be presented by a single point (ρ, θ) in Hough space (Figure 1). The Kikuchi bands appear as bright regions or peaks in Hough space, which are easily detected and used to calculate the original band positions (Figure 2).

## Interpreting the Diffraction Pattern

The centre lines of the Kikuchi bands correspond to where the diffracting planes intersect with the phosphor screen. Hence, each Kikuchi band can be indexed by the Miller indices of the diffracting crystal plane which formed it. The intersections of the Kikuchi bands correspond to zone axes in the crystal.

The pattern is a gnomonic projection of the diffracted cones of electrons onto the phosphor screen. The semi-angle of the diffracted cones of electrons is (90 - θ)^{o}. For EBSD, this is a large angle so the Kikuchi bands approximate to straight lines. For example, the wavelength of 20 kV electrons is 0.00859 nm and the spacing of the (111) plane in Aluminium is 0.233 nm, making the cone semi-angle 88.9°.

The width w of the Kikuchi bands close to the pattern centre is given by:

where l is the distance from the sample to the screen. Planes with wide d-spacings give thinner Kikuchi bands than those with narrower planes. Because the diffraction pattern is bound to the crystal structure of the sample, as the crystal orientation changes, the resultant diffraction pattern also changes. The positions of the Kikuchi bands can therefore be used to calculate the orientation of the diffracting crystal (Figure 1).

This animation shows the relationship between the crystal orientation and simulated diffraction pattern for a face centred cubic crystal. As the crystal rotates the diffraction pattern moves. (The simulated diffraction pattern is from a sample tilted 70° to the horizontal and the crystal orientation is viewed along the direction perpendicular to the sample).

## Calibrating an EBSD system

An EBSD system must be calibrated to work correctly. This typically, involves measuring the sample-to-screen distance and the pattern-centre position on the phosphor screen, (the pattern centre is the point on the screen closest to the generation point of the diffraction pattern on the sample).

There are several methods for calibrating an EBSD system. The preferred method is to collect diffraction patterns from the same point on a sample with the detector placed at different insertion positions (Figure 1) and at different working distance. Using image correlation, it is then possible to identify the pattern centre at these different geometries and be measuring the zoom factors between the images it is possible to calculate the distance to the sample.

In a modern EBSD system this process is automated so that patterns are collected at these different geometries and a matrix is created, so that as the system is used it is always in calibration, what ever the geometry conditions.